# Lemniscate of Bernoulli as Locus in Complex Plane

## Theorem

The locus of $z$ on the complex plane such that:

$\cmod {z - a} \cmod {z + a} = a^2$

## Proof

 $\ds \cmod {z - a} \cmod {z + a}$ $=$ $\ds a^2$ $\ds \leadsto \ \$ $\ds \paren {\paren {x - a}^2 + y^2} \paren {\paren {x + a}^2 + y^2}$ $=$ $\ds a^4$ Definition of Complex Modulus $\ds \leadsto \ \$ $\ds \paren {x^2 - 2 a x + a^2 + y^2} \paren {x^2 + 2 a x + a^2 + y^2}$ $=$ $\ds a^4$ $\ds \leadsto \ \$ $\ds \paren {\paren {x^2 + a^2 + y^2} - 2 a x} \paren {\paren {x^2 + a^2 + y^2} + 2 a x}$ $=$ $\ds a^4$ $\ds \leadsto \ \$ $\ds \paren {x^2 + a^2 + y^2}^2 - 4 a^2 x^2$ $=$ $\ds a^4$ Difference of Two Squares $\ds \leadsto \ \$ $\ds \paren {x^2 + y^2}^2 + a^4 + 2 \paren {x^2 + y^2} a^2 - 4 a^2 x^2$ $=$ $\ds a^4$ $\ds \leadsto \ \$ $\ds \paren {x^2 + y^2}^2 + 2 a^2 \paren {x^2 - 2 x^2 + y^2}$ $=$ $\ds 0$ $\ds \leadsto \ \$ $\ds \paren {x^2 + y^2}^2 + 2 a^2 \paren {y^2 - x^2}$ $=$ $\ds 0$ $\ds \leadsto \ \$ $\ds \paren {x^2 + y^2}^2$ $=$ $\ds 2 a^2 \paren {x^2 - y^2}$

which is the equation defining a lemniscate of Bernoulli with major semiaxis $a \sqrt 2$.

$\blacksquare$